matrix models and the quantum structure of space, time and...

Matrix models and the quantum structure ofspace, time and matter

Harold Steinacker

Department of Physics, University of Vienna

February 2, 2018

H. Steinacker Matrix models and the quantum structure of space, time and matter

Motivation Quantum geometry IKKT model, NC branes Cosmological space-times

outline:

status of “fundamental theory”

quantum space-time & geometry

matrix models

4D quantum spaces & cosmological space-times

towards particle physics, outlook



present understanding of fundamental matter & interactions:

standard model of elementary particle physics= quantum field theory

governs fundamental constituents of matter & interactionsexcept for gravity!

general relativity (GR)= classical geometrical theory of gravity

however: class. Einstein equations inconsistent with QM:

Rµν −12

gµνR+ Λgµν︸︷︷︸classical

?=

8πGc4 Tµν︸︷︷︸

quantum

GR resists quantization, not renormalizable



underlying issue: superposition principle of quantum mechanics

superposition of massive objects

⇓

superposition of “gravitational potentials”

⇒ need quantum theory of geometry & gravity

requires new framework for quantum space-time



Observational issuepresent cosmological concordance model: ΛCDM model

95 % of the Universe is not understood !!either

General Relativity is wrong, or

particle physics is incomplete, or

both (most likely)

→ expect major change in fundamental physics !H. Steinacker Matrix models and the quantum structure of space, time and matter


approaches to quantum & gravity

direct quantization of (pure) gravity:

loop quantum gravity, canonical QGcausal dynamical triangulationsasymptotic safety... (all have problems!)

quantization of very special models which include gravity

string theory (landscape? def?)Matrix Models (this talk!)

(holography)

“quantum” (noncommutative) geometrydescribed by Matrix Modelsphysics emerges from fluctuations



fuzzyness of space-time

Q.M. & G.R.⇒ break-down of classical space-time

measure object of size ∆x

Q.M. ⇒ energy E ≥ ~kc ∼ ~c∆x

G.R. ⇒ ∆x ≥ RSchwarzschild ∼ GMc2 ≥ ~G

c3∆x

⇒ (∆x)2 ≥ G~c3 = L2

Pl , LPl = 10−33cm

classical geometry breaks down, need pre-geometric framework

“fuzzy”, “foam-like” structure of space-time J. Wheeler 1955

“Unanwendbarkeit der Geometrie im Kleinen” Schrodinger 1934

why worry? because

virtual quantum effects in QFT probe UV scale, very significant!!(cf. UV divergences in 4D QFT)



what is the quantum structure of geometry?

... lots of possibilities, generic problem:

modifications of geometry in UV → uncontrollable, unexpectedeffects in IR (cf. UV divergences in 4D QFT)

most approaches will fail, need protection mechanism

more precise statement: Doplicher Fredenhagen Roberts 1995

∑∆xµ∆xν ≥ L2

Pl

... space-time uncertainty relations, follows from

[Xµ,X ν ] = iθµν (cf. Q.M. !)

... noncommutative (quantum) space-time



quantized space-time & geometry

old idea: Schrodinger, Heisenberg, Snyder, ...

borrow idea from QM:

QM = quantization of phase space [X ,P] = i~

quantum space = quantization [Xµ,X ν ] = iθµν of space-time

inherent uncertainty of space(time)use Q.M. techniques (coherent states, ...) for spacetime & geometry



basic message of GR:

cannot separate space-time↔ matter

need unified model governing space-time, matter & interaction

task:

find correct model which governs dynamical quantumspace-times

(many approaches conceivable, most will fail)

find correct space-time solutions

identify mechanisms for emergent physics(gauge theory & gravity)

how to choose model & framework ?



two crucial insights:

insight 1 (around 2000):

Yang-Mills gauge theory arises from fluctuations of quantum spacesin Matrix Models

[Xµ,X ν ] = iθµν , Xµ → Xµ +Aµ

[Xµ +Aµ,X ν +Aν ] = iθµν + i ∂µAν − ∂νAµ + i[Aµ,Aν ]︸︷︷︸Fµν

simpler than in classical space-time !! (use [Xµ,A] ∼ iθµν∂νA)

S = Tr [Xµ,X ν ][Xµ,Xν ] ∼∫

FµνFµν + c

what about gravity?

insight 2:

Matrix Models S = Tr [Xµ,X ν ][Xµ,Xν ] → dynamical quantum spaces



useful guideline: Schomerus, Chu-Ho, Seiberg-Witten, ... ∼ 1999

NC spaces & gauge theory arise on D-branes in string theory... “NonCommutative (quantum) field theory”

problem: UV/IR mixing Minwalla, van Raamsdonk, Seiberg 1999

virtual quantum effects in QFT probe shortest scalesstring-like behavior |x〉〈y | at high energies, non-localUV divergences in QFT new IR divergences

→ naive approaches will fail

avoided in one special model: IKKT matrix model2-fold origin/interpretation:

maximally SUSY Yang-Mills theorynonperturbative string theory



Matrix Models as candidates for a fundamental theory

IKKT or IIB model Ishibashi, Kawai, Kitazawa, Tsuchiya 1996

S[X ,Ψ] = −Tr(

[X a,X b][X a′,X b′

]ηaa′ηbb′ + Ψγa[X a,Ψ])

X a = X a† ∈ Mat(N,C) , a = 0, ...,9, N large

gauge symmetry X a → UX aU−1, SO(9,1), SUSY

proposed as non-perturbative definition of IIB string theory

maximally SUSY Yang-Mills in 3+1 dim.

quantization via matrix “path integral”

Z =∫

dXdΨ eiS[X ]

cf. BFSS model 1996: matrix quantum mechanics (class. time)



leads to “matrix geometry” (≈ NC geometry):

SE ∼ Tr [X a,X b]2 ⇒ config’s with small [X a,X b] 6= 0 dominate

i.e. “almost-commutative” configurations

∃ quasi-coherent states |x〉, minimize∑

a〈x |∆X 2a |x〉

(X a) ≈ diag., spectrum =: M ⊂ R10

〈x |X a|x ′〉 ≈ δ(x − x ′)xa, x ∈M

“condensation” of matrices→ geometry:

NC branes embedded in target space R10

X a ∼ xa : M ↪→ R10

matrices = observables X a = quantized functions xa ( cf. Q.M)



examples of matrix geometries:

three 2× 2 matrices

X 3 =

(x(1)

x(2)

)= x(1)|1〉〈1|+ x(2)|2〉〈2|,

X 1 + iX 2 =

(x(12)

)= x(12)|2〉〈1|

X 1 − iX 2 =

(x(21)

)= x(21)|1〉〈2|

describe two points at x(1), x(2) ∈ RD

• • (“point branes”)off-diagonal matrices ≈ strings connecting branes

spectrum of X a ↔ location in RD

→ minimal fuzzy sphere S2 ↪→ R3

X a = σa, X 21 + X 2

2 + X 23 = 3

4



examples of matrix geometries:

three 2× 2 matrices

X 3 =

(x(1)

x(2)

)= x(1)|1〉〈1|+ x(2)|2〉〈2|,

X 1 + iX 2 =

(x(12)

)= x(12)|2〉〈1|

X 1 − iX 2 =

(x(21)

)= x(21)|1〉〈2|

describe two points at x(1), x(2) ∈ RD

• ((66 • (“point branes”)

off-diagonal matrices ≈ strings connecting branes

spectrum of X a ↔ location in RD

→ minimal fuzzy sphere S2 ↪→ R3

X a = σa, X 21 + X 2

2 + X 23 = 3

4



generic class of matrix geometries:

fuzzy spaces = quantized Poisson manifolds (cf. phase space, QM)

X a ∼ xa : M ↪→ RD

quantization map:

Q : C(M) → End(H) ∼= Mat(N,C) H = CN

such that

Q(f )Q(g) = Q(fg) + O(θ), θ ∼ 1N

[Q(f ),Q(g)] = Q(i{f ,g}) + O(θ2)

f = f (X ) = f † ∈ Mat(N,C) ... quantized algebra of functions onM(“observables”) cf. QM!

in particular:X a = Q(xa)

typically Tr Q(f ) ∼∫M f (x)



Example: the fuzzy sphere S2N (Hoppe, Madore)

classical S2 : xa : S2 ↪→ R3

(x1)2 + (x2)2 + (x3)2 = 1

fuzzy sphere S2N :

3 matrices X a := 1√CNπN(Ja) ... spin N−1

2 generators of su(2)

(X 1)2 + (X 2)2 + (X 3)2 = 1l,

[X a,X b] = i√CNεabc X c ∼ i

N , CN = 14 (N2 − 1)

algebra A = Mat(N,C) ... quantized functions on S2N

S2N ... quantization of S2 with Poisson bracket {xa, xb} = 2

N εabc xc



covariance: SO(3) rotations

so(3)×A → A(Ja, φ) 7→ [πN(Ja), φ]

→ fuzzy spherical harmonics, UV cutoff

A = Mat(N,C) ∼= (N)⊗ (N) = (1) ⊕ (3) ⊕ ...⊕ (2N − 1)

= {Y 00 } ⊕ {Y 1

m} ⊕ ...⊕ {Y N−1m }.

S2N is fully covariant!

metric encoded in NC Laplace operator

2 : A → A,2φ = [X a, [X b, φ]]δab = 1

CNJaJaφ

compute 2Y lm = 1

CNl(l + 1)Y l

m

spectrum identical with classical case ∆gφ = 1√|g|∂µ(√|g|gµν∂νφ)

⇒ effective metric of 2 = round metric on S2



can measure such matrix geometries {X a}:

idea: Berenstein - Dzienkowski arXiv:1204.2788, Ishiki arXiv:1503.01230,Schneiderbauer - HS arXiv:1606.00646

measure energy E(x) of string connectingM with point at x ∈ RD

location ofM⊂ RD ↔ minima of E(x)

Mathematica package “Bprobe” DOI 10.5281/zenodo.45045Schneiderbauer - HS

examples:

squashed fuzzy CP2N ⊂ R6 fuzzy torus T 2

N



4D covariant quantum spaces

issue: NC spaces: [Xµ,X ν ] =: iθµν 6= 0breaks Lorentz invariance

however: ∃ fully SO(5) covariant fuzzy four-sphere S4N

Grosse-Klimcik-Presnajder 1996; Castelino-Lee-Taylor; Ramgoolam; Kimura;

Hasebe; Medina-O’Connor; Karabail-Nair; Zhang-Hu 2001 (QHE!) ...

complication/bonus: “internal structure”

fluctuations includes spin 2 modes → higher spin theory

“gravity“ naturally emergesHS arXiv:1606.00769 M. Sperling, HS arXiv:1707.00885

Euclidean case unphysical



∃ analogous

covariant cosmological quantum space-time solutions of IKKT modelsHS arXiv:1710.11495, arXiv:1709.10480

exactly homogeneous & isotropic

finite density of microstates

mechanism for Big Bang



starting point: Euclidean fuzzy hyperboloid H4n

Hasebe arXiv:1207.1968Mab ... hermitian generators of so(4,2),

[Mab,Mcd ] = i(ηacMbd − ηadMbc − ηbcMad + ηbdMac) .

ηab = diag(−1,1,1,1,1,−1)choose “short” discrete unitary irreps Hn (“minireps”, doubletons)

special properties:

irreducible under so(4,1), multiplicity-free

positive discrete spectrum

spec(M05) = {E0,E0 + 1, ...}, E0 = 1 +n2

eigenspaces ofM05 finite-dim.



fuzzy hyperboloid H4n :

def. 5 hermitian matrices

X a := rMa5, a = 0, ...,4

satisfy

ηabX aX b = X iX i − X 0X 0 = −R21l, R2 = r2(n2 − 4)

[X a,X b] = ir2Mab =: iΘab

one-sided hyperboloid in R1,4, covariant under SO(4,1)

note: induced metric is EuclideanH. Steinacker Matrix models and the quantum structure of space, time and matter


oscillator construction: 4 bosonic oscillators [ψα, ψβ] = δβα

thenX a = r ψγaψ

on Fock space Hn γa ... SO(4,1) Gamma matrices

H4n is really quantized CP1,2 = S2 bundle over H4, selfdual θµν

(analogous to S4N )

spec(X 0 =M05) discrete, finite degeneracy

⇒ finite density of microstates!



open FRW universe from projected H4n HS arXiv:1710.11495

Yµ := Xµ, for µ = 0,1,2,3 (drop X 4 !)

= projection of H4n in R1,3 via

Yµ ∼ yµ : CP1,2 → H4 Π−→ R1,3 .

is solution of IKKT with m2 = −3r2, since

[Yµ, [Yµ,Y ν ]] = ir2[Yµ,Mµν ] (no sum)

= r2

Y ν , ν 6= µ 6= 0−Y ν , ν 6= µ = 00, ν = µ

hence

2Y Yµ = [Y ν , [Yν ,Yµ]] = 3r2Yµ (= eom of IKKT)



properties:

SO(3,1) manifest ⇒ foliation into SO(3,1)-invariant space-like3-hyperboloids H3

t ( homogeneous & isotropic! )

double-covered FRW space-time with hyperbolic (k = −1)spatial geometries

ds2 = dt2 − a(t)2dΣ2,

dΣ2 ... SO(3,1)-invariant metric on space-like H3



effective metric: H.S. arXiv:1003.4134

encoded in 2Y = [Yµ, [Yµ, .]] ∼ 1√|G|∂µ(√|G|Gµν∂ν .):

Gµν = α gµ′ν′ [θµ′µθν

′ν ]S2 , α =√|θµν ||γµν |

∼= α diag(c0(η), c(η), c(η), c(η))

where [.]S2 ... averaging over the internal S2

c(η) = 1− 13 cosh2(η)

c0(η) = cosh2(η)− 1 ≥ 0

signature change at c(η) = 0

cosh2(η0) = 3 ...Big Bang!

Euclidean for η < η0, Minkowski (+−−−) for η > η0



FRW metric and scale factor

ds2G = dt2 − a2(t)dΣ2

withdy0

dt=

c0(y0)12

|c(y0)| 34, a(t) = |c(y0)| 12 y2

0 .

Big Bang:

shortly after the BB :

a(t) ∝ c(t)14 ∝ t1/7

1 2 3 4 5

0.5

1.0

1.5

2.0

a(t)

conformal factor & 4-volume form |θµν | responsible for singularexpansion!



late times:

linear coasting cosmology

a(t) ∼ 3√

32

t .



a(t) ∼ t is remarkably close to observation:

age of univ. 13.9× 109y from present Hubble parametersimilar to Milne Univ;

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5

Sca

le f

acto

r -

a(t

)

t/t0

a(t) - ΛCDMa(t) - Milne Universe

artificial within GR, natural in M.M.gravity should emerge below cosm. scales (?!)

can reproduce SN1a (without acceleration)

different physics for early universe (CMB etc.)

cf. Melia Mon.Not.Roy.Astron.Soc. 446 (2015); Benoit-Levy , ChardinAstron.Astrophys. 537 (2012); Nielsen, Guffanti, Sarkar Sci.Rep. 6(2016)



other features:

∃ Euclidean pre-BB era

2 sheets with opposite intrinsic“chirality”

fluctuations on internal S2 → higher-spin fluctuation modes

Aµ = θµνhνρ(x)Pρ

expect (higher-spin-extended) gravity (spin 2 = gravity!)in progress, cf. HS arXiv:1606.00769 M. Sperling, HS arXiv:1707.00885



towards particle physics from the IKKT model

IKKT model has 9+1 “dimensions“

idea (cf. string theory):

consider solutionsM3,1 ×KN with 6 fuzzy extra dimensions

requires embedding of standard model fields in adjoint of SU(N):indeed: A. Chatzistavrakidis, H.S., G. Zoupanos arXiv:1107.0265

H. Grosse, F. Lizzi H. S. arXiv:1001.2703

Ψ =

02 0 0 lL QL

0(

0 eR0 νR

)QR

0 003

,

whereQL =

(uLdL

), lL =

(νLeL

), QR =

(dRuR

)similar to brane constructions of the S.M. in string theory



3 generations

from projected quantized coadjoint SU(3) orbits in extra dim.Ya = πµ(Ta), Ta ... su(3) root generators

C[µ] ↪→ R8 Π→ R6

(ya)a=1,...,8 7→ (ya)a=1,2,4,5,6,7

= solutions of IKKT model with cubic term,4- or 6-dimensional self-intersecting variety in R6

H.S., J. Zahn arxiv:1409.1440, H.S. arXiv:1504.05703chiral fermions from strings linking self-intersecting sheets

ψα,Λ = |µ′〉〈µ| , |µ〉 ... coherent statestriple self-intersection→ 3 generations!



no obstacle in principle to get (extended) standard model from IKKT,

no need for string compactifications

the simplest possible model might actually work !?



Summary

understanding of fundamental physics is not complete↪→ opportunities!

matrix models: promising framework for quantum theory ofspace-time & geometry

analogous math as in Q.M.

extremely simple, good UV behavior (IKKT model)

all ingredients for gravity & partice physics

cosmological space-times & BB from fuzzy H4n

finite d.o.f. per volume

gravity expected to emerge

... exciting potential, seize the opportunity!


matrix models and the quantum structure of space, time and...

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